Centralizing generalized derivations on polynomials in prime rings. (Q1758315)

Let \(R\) be prime ring with extended centroid \(C\), let \(I\) be a nonzero right ideal of \(R\), and let \(f(x_1,\dots,x_n)\) be a nonzero noncommutative polynomial over \(C\). The main result describes all possible situations when a generalized derivation \(G\) of \(R\) satisfies \([G(f(r_1,\dots,r_n)),f(r_1,\dots,r_n)]\in C\) for all \(r_i\in R\). The case where \(f\) is multilinear and \([G(f(r_1,\dots,r_n)),f(r_1,\dots,r_n)]=0\) was considered earlier by the same author [in Isr. J. Math. 162, 93-108 (2007; Zbl 1142.16023)].